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Spectral vs Energy Efficiency in 6G:
Impact of the Receiver Front-End
Angel Lozano, Fellow, IEEE, Sundeep Rangan, Fellow, IEEE
Abstract—This article puts the spotlight on the receiver
front-end (RFE), an integral part of any wireless device
that information theory typically idealizes into a mere
addition of noise. While this idealization was sound in the
past, as operating frequencies, bandwidths, and antenna
counts rise, a soaring amount of power is required for the
RFE to behave accordingly. Containing this surge in power
expenditure exposes a harsher behavior on the part of the
RFE (more noise, nonlinearities, and coarse quantization),
setting up a tradeoff between the spectral efficiency under
such nonidealities and the efficiency in the use of energy
by the RFE. With the urge for radically better power
consumptions and energy efficiencies in 6G, this emerges
as an issue on which information theory can cast light at a
fundamental level. More broadly, this article advocates the
interest of having information theory embrace the device
power consumption in its analyses. In turn, this calls for
new models and abstractions such as the ones herein put
together for the RFE, and for a more holistic perspective.
I. MOTI VATI ON
Energy efficiency was an important driver in the design
of 1G and 2G wireless standards where, for instance,
more efficient amplifiers were enabled by the adoption of
(respectively analog and digital) signaling formats tolerant
of nonlinear amplification. By the time 3G came to be,
however, the perception of spectrum scarcity in the low
frequency bands had brought about a shift in priorities,
and energy efficiency has since taken a back seat. Indeed,
a chief thrust in 3G, 4G, and 5G, was to improve the
spectral efficiency.
In the post-5G era, the pendulum is swinging back—
not so much because the demand for spectral efficiency
is relenting, but because energy efficiency is becoming
an imperative. The ICT sector is anticipated to devour
a staggering 20% of the global electricity production by
2030 and, homing in on wireless networks, the radio
access is the most energy-hungry portion [1]. Meanwhile,
at the device end, autonomy and battery life are expected
to acquire paramount importance. This mounting pressure
on the use of energy brings renewed interest in the tradeoff
between spectral and energy efficiency, for years skewed
all the way towards the former.
A. Lozano is with Universitat Pompeu Fabra, 08018 Barcelona (e-
mail: angel.lozano@upf.edu). His work is supported by the
Fractus-UPF Chair on Tech Transfer and 6G, and by the ICREA
Academia program. S. Rangan is with New York University, Brooklyn,
BY (e-mail: srangan@nyu.edu). His work is supported in part by
NSF grants 1952180, 2133662, 2236097, 2148293, and 1925079, along
with the industrial affilates of NYU Wireless.
Bit rate
Power
1 Mbps 10 100 1 Gbps 10
1 mW
10
100
1 W
10
New 6G regime
Ex: 20 m range
∼1 mW TX
∼1 mW RX
160 Mbps
Bluetooth
low energy
Sub-6 GHz
cellular
MmWave
cellular
MmWave
WiFi
Fig. 1: Power and rate regions of operation of existing wireless tech-
nologies, alongside the new low power, high bit rate regime of potential
interest for 6G. An example in this regime is pinpointed based on
the models in this paper combined with current RF device power vs.
performance characteristics (see Appendix A for details).
At the same time that they regain importance, energy ef-
ficiency assessments need to become more holistic. Clas-
sically, the only power component that information theory
has concerned itself with is the transmit power, but with
the progression towards ever higher carrier frequencies,
much broader bandwidths, and multiplied antenna counts,
the power consumed by the circuitry is bound to become
comparable to the transmit power at the infrastructure end
[1, Fig. 1] and might outright dwarf it at many devices.
By far the biggest contributor to the circuit power
consumption is certain to be the receiver front-end (RFE),
whose consumption is swelling already in 5G. For 6G,
carrier frequencies could reach 300 GHz, with multi-
gigahertz bandwidths and on the order of 64 antennas
at mobile units [1], posing a major challenge. Even for
lower frequencies, bandwidths, and antenna numbers, the
RFE consumption becomes an issue if high autonomies
and/or miniature batteries are desired. As illustrated in
Fig. 1, in contrast with existing paradigms of high bit
rates with a high transmit power or low bit rates with a
low transmit power, many 6G devices may be in a new
class of their own: high bit rates over short distances with
a low transmit power. In such devices, the RFE power
consumption might overshadow the transmit power, and
lowering the former would enable a powerful regimes
of use cases for applications such as smart wearables,
untethered cameras, virtual reality goggles, connectivity
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content may change prior to final publication. Citation information: DOI 10.1109/MBITS.2023.3322975
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2
modules, or compact short-range access points devoid of
cooling systems.
Crucially, any effort to tame the power expended by
an RFE pushes it away from ideality, aggravating its
imperfections: higher noise floor, nonlinear behavior, and
coarser quantization [2]. Characterizing the impact of the
RFE on the fundamental performance of a communica-
tion system requires models that abstract these essential
aspects of noisiness, nonlinearity, and quantization coarse-
ness in a manner that is useful from an information-
theoretic vantage, as well as a physically-based model for
the concomitant power consumption. Coarse quantization
has been dealt with extensively already, and important
results have emerged (see [3], [4] and references therein).
This article expands the modeling scope to also subsume
the other aspects.
Armed with a model for the RFE, both the spectral
efficiency that it enables (in bits/s/Hz) and its energy
efficiency (in bits/s/Watt or bits/Joule) can be character-
ized, with the latter also expressible through its reciprocal,
the energy per bit (in Joules/bit). The spectral efficiency
and the RFE energy per bit inform, respectively, of the
utilization of the bandwidth and of the energy stored by
the device’s battery. These two performance measures
are informative on their own, yet they are even more
revealing when pitted against each other. This amounts
to expressing the spectral efficiency, not as a function
of the energy expended by the RFE per unit of time,
but per unit of information. Indeed, pushing the RFE
towards ideality increases its power consumption, but that
may be worthwhile if sufficiently more bits can then be
sent through. Conversely, relaxing the RFE specifications
lowers its consumption, but that need not be fruitful
from an energy per bit standpoint, depending on how
much the bit rate drops. A spectral-vs-energy efficiency
assessment is the appropriate framework to make these
determinations, examine the interplay of the RFE key
knobs, and glean design guidelines that explicitly account
for the energy cost of concealing the RFE’s imperfections.
These RFE imperfections, which motivate the analysis
in the first place, also complicate it, as they give rise to
settings that deviate from the familiar linear channel with
additive white Gaussian noise (AWGN). These broader
settings are described, relevant works are surveyed, and
open issues are identified.
Altogether, this article contemplates the impact of the
RFE in potential 6G designs operating at higher carrier
frequencies, much wider bandwidths, increased antenna
counts, and/or with far lower power consumption limits
and energy budgets. Looking beyond, the formulation
could be augmented with other expanding contributions
to the power consumption (say the channel decoder),
for an ever more comprehensive assessment of the cost
of operating a receiver. This could then be conceivably
blended with its transmitter counterpart. Indeed, in terms
of efficiency what matters is the total expended energy
[5], [6]; holistic assessments align with 3GPP’s energy
TX Wireless
Channel RFE Baseband
z
x r y
RX
Channel DecoderEncoder
Fig. 2: Transmission chain with transmitter, channel, and receiver. From
the information-theoretic vantage of encoding and decoding, the RFE
can be subsumed into the channel.
consumption and efficiency metrics, which account for the
total energy irrespective of where it is consumed.
For proofs of some of the results and further consider-
ations, readers are referred to [7].
II. RFE MOD EL
For starters, let us consider the setting in Fig. 2, with a
single-antenna receiver and bandwidth B. The discrete-
time complex baseband transmit signal is xwhile the
noiseless received signal is r=hx for a given complex
channel gain, h. The energies per symbol are Ex=E[|x|2]
and Er=E[|r|2] = E[|h|2]Ex.
A. Linear RFE Model
A wireless receiver consists of two stages: the RFE,
which effects the downconversion, filtering, and digital-
ization, and the baseband processor, which demodulates
and decodes. The standard model for the RFE is
y=r+z, (1)
where zis complex Gaussian noise with variance kT F ;
here, kT is the minimum theoretical value (−174 dBm/Hz
at room temperature) while F > 1is the noise figure
quantifying the increase in noise due to RFE nonideality.
With that, the signal-to-noise ratio is SNR =SNRideal/F
given
SNRideal =Er
kT .(2)
This AWGN setting has spawned much of the wisdom
on the fundamental limits of reliable communication. In
particular, the highest achievable bit rate is
R=Blog2(1 + SNR),(3)
attained when xis complex Gaussian. The corresponding
spectral efficiency is C=R/B.
Having the RFE approach the behavior in (1) is how-
ever very costly in terms of power consumption and, as
advanced, as one attempts to lower that cost, the behavior
becomes less benign. This less benign functioning can be
captured by the broader model y= Φ(r, z), where Φ(·)
is a memoryless, generally nonlinear function.
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3
+
r
z
S(·)× Qb(·)y
z√α
u
Saturation ADCGain
Noise
Fig. 3: RFE modeled as a cascade of four operations: additive noise,
saturation, gain, and quantization.
B. RFE Operations
The specific form for Φ(·)propounded here is, as
illustrated in Fig. 3,
Φ(r, z) = Qb√αS(r+z),(4)
which expresses a cascade of components, each a critical
feature of the RFE as detailed next.
•Additive noise: As in the linear model, zis the noise,
with variance kT F .
•Saturation. The transformation S(·)brings into the
model the finite linear range via
S(ξ) = pEmax ϕ|ξ|
√Emax ξ
|ξ|,(5)
where ϕ(·)∈(0,1] is a function satisfying ϕ(0) = 0
and ϕ′(0) = 1. For the sake of analysis, two concrete
such functions can be ϕ(·) = tanh(·), which is
smooth, and ϕ(·) = min(·,1), an outright clipping.
The transform behaves linearly for small inputs, but
it saturates at a signal power of Pmax =EmaxB.
•Quantization. The function Qb(·)models the analog-
to-digital converter (ADC), with bbits per each of the
in-phase and quadrature dimensions.
•Gain control. Critical to proper operation of the
ADC is a scaling of its input to prevent overflow
or underflow. This is represented by α.
Altogether, the RFE is characterized by the knobs
(F, Pmax, b). In addition, the performance is influenced
by the strategy for setting the gain control.
C. Power Consumption
There is considerable range in the RFE power consump-
tion for a set of performance attributes, with deviations
due to the process technology, form factor, and other
considerations. For the purpose here, what is sought is
not a model to make precise consumption predictions, but
to—based on the scaling laws of circuits and devices—
abstract the dependencies on the RFE knobs and on the
system parameters, chiefly frequency and bandwidth.
The noise figure is determined primarily by the low-
noise amplifier (LNA), at whose input the signal is weak-
est. The most widely used scaling for the LNA power
consumption is with fc/(F−1) where fcis the carrier
frequency [8]; this is the scaling adopted here, even if
[9] suggests that, at very low powers, the scaling may
be more aggressive, namely with f2
c/(F−1)3. As of the
saturation, it occurs in both the LNA and mixer, and the
Figure of Merit Units Value Remark
γADC fJ/qt 165 Based on ADC data [14]
γNF fJ 140 Based on LNA data [9]
γmax -5000 Based on mixer data [2],
[15], [16]
TABLE I: Figures of merit based on recent device and circuit surveys.
incurred power is reasonably modeled as proportional to
Pmax [10]–[13]. With no single accepted rule for how the
noise figure and the saturation power consumptions should
be combined [11], [13], this text espouses their addition;
this conservative choice is sure to be valid for moderate
perturbations around nominal values of Fand Pmax.
Then, the two ADCs required to process a complex
signal consume a power that scales with Bκbwhere κ≈4
for signal-to-quantization ratios in excess of 40–50 dB
while κ≈2for the lower ratios at which wireless systems
operate [14]; κ= 2 is thus considered in the sequel.
Although, beyond a few hundred megahertz, the ADC
consumption would become quadratic in the bandwidth,
the linear scaling can always be retained through par-
allelization and ADC pipelining (multiple low-resolution
stages cascaded to obtain a higher resolution). Only if it
were truly necessary to quantize a single band exceeding
hundreds of megahertz would a quadratic behavior be
experienced.
The gain control, finally, is typically performed at
baseband and consumes negligible power relative to the
rest. All in all, the proposed model for the RFE power
consumption is
PRFE =γNF
fc
F−1
|{z }
noise figure
+γmaxPmax
| {z }
saturation
+γADCB2b
| {z }
ADC
(6)
with indicative values for the figures of merit γADC,γNF, and
γmax, presented in Table I . In particular, γADC improves by
about 1.5dB per year, with a fundamental limit imposed
by physics anticipated at γADC ≈0.1fJ/qt [14]. See [7]
for more details on the figures of merit.
Interestingly, the power consumption term induced by
each knob involves a distinct system parameter:
•The power that must be burned to attain a certain
noise figure is determined by the carrier frequency,
but neither the bandwidth nor the received signal
strength. And, rewriting it as
γNFfc
F−1=γNFfc
SNRideal
SNR −1,(7)
it cleanly connects the consumed power, the carrier
frequency, and the SNR degradation.
•The power that needs to be expended to stretch the
range of unsaturated signals depends on the received
signal strength, and hence it relates to SNRideal.
•The power consumed to operate at a certain ADC
resolution is contingent on the bandwidth, but not on
the carrier frequency or the received signal strength.
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content may change prior to final publication. Citation information: DOI 10.1109/MBITS.2023.3322975
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4
III. SPE CT RA L AN D ENERGY EFFIC IENCY
When a nonlinear transformation sits between encoder
and decoder, the noise is compounded by distortion and
quantifying the information-theoretic performance limits,
as well as strategies that can approach them, becomes
notoriously difficult. In such a context, a very useful result
can be formulated on the basis of the signal-to-noise-and-
distortion ratio
SNDR =ρ2
xy
1−ρ2
xy
(8)
where noise and distortion are blended via the squared
correlation coefficient
ρ2
xy =|E[x∗y]|2
ExEy
.(9)
Based on the SNDR, the spectral efficiency with complex
Gaussian signaling satisfies [17]
C≥log2(1 + SNDR).(10)
This bound can be obtained from the Bussgang-Rowe
decomposition, whereby the output of a nonlinear trans-
formation is split into a scaled version of the input
plus uncorrelated distortion (in general neither Gaussian
nor independent of the input); the bound arises by then
replacing that distortion with Gaussian noise of the same
variance.
When a unitary transformation is applied between the
RFE and the decoder, for instance the time-frequency
transformation in multicarrier signaling, the distortion is
thrown around and asymptotically (in the dimensions of
the unitary transformation, say the number of subcarriers)
it is rendered Gaussian; then, the bound gives the actual
achievable spectral efficiency [17]. Without a post-RFE
transformation, the distortion remains signal-dependent
and non-Gaussian; these attributes could conceivably be
taken advantage of by the decoder, rendering the lower
bound somewhat conservative. Customarily though, the
decoder treats the distortion as additional Gaussian noise,
whereby its effect (on Gaussian signals) is indeed that of
additional Gaussian noise [18], [19]. Motivated by this
argument, the lower bound in (10) is regarded as the
achievable spectral efficiency in the sequel.
As of the RFE’s energy efficiency, it equals R/PRFE. At
the infrastructure end, this is often measured in Mb/kWh,
reflecting that electricity is billed in kWh, while at mobile
devices b/J is a more fitting unit. More germane to
information theory is actually its reciprocal, the RFE
energy per bit
ERFE
b=PRFE
R=1
CγNF
F−1
fc
B+γmaxEmax +γADC 2b
(11)
in J/b; this is the measure adopted henceforth. Note that
we have used the fact that Pmax =EmaxB. Importantly, ERFE
b,
which relates to the RFE power consumption, should not
be confused with the transmit and received energy per bit,
both associated with the radiated power [20, Sec. 4.2].
SDR [dB]
0
15
20
25
30
10
5
Saturation backoff [dB]
0 10 20 30-5 5 15 25
b=1
b=2
b=3
b=4
b=5
b=6
Fig. 4: SDR as a function of the ADC resolution and the saturation
backoff.
An important implication of (11) is that the dependence
of the RFE energy per bit on the bandwidth and carrier
frequency is only through the ratio, B/fc.
A. A Closer Look at the SNDR
Without loss of generality, the ADC can be designed
for a unit-energy input, whereby the required gain control
is
α=1
E[|S(r+z)|2].(12)
With this gain, and given how x,r, and zare related
through SNRideal, (8) can be manipulated (See [7]) into
SNDR =SNRideal ρ2(b, ν)
F+SNRideal (1 −ρ2(b, ν)) ,(13)
where
ρ2(b, ν) = |E[(r+z)∗y]|2
(Er+N0)Ey
(14)
implicitly depends on the function ϕ(·)in (5) and must
generally be computed numerically. Interestingly, ρ(b, ν)
depends only the ADC resolution and on the saturation
backoff, the latter given by
ν=Emax
Er+N0
.(15)
No matter how high SNRideal, the SNDR is curbed, namely
SNDR ≤ρ2(b, ν)
1−ρ2(b, ν)(16)
whose right-hand side is the noiseless signal-to-distortion
ratio (SDR). Fig. 4 shows this SDR for ϕ(·) = tanh(·),
illustrating the effects of the resolution and the saturation
backoff.
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5
B. Only Noise and Quantization
When the backoff is large, the input is always below the
saturation level and the distortion is caused only by the
ADC. Then, the gain control becomes α= 1/(Er+N0)
and (13) continues to hold, only with ρ(·)a function of
solely the resolution, namely
ρ2(b) = |E[u∗Qb(u)] |2
E[|Qb(u)|2](17)
where u∼ NC(0,1) represents the gain-controlled signal
being fed to the ADC. Since ρ(b)∈[0,1],
0≤SNDR ≤SNRideal
F(18)
as one would expect, with the highest value corresponding
to infinite resolution. At low SNRideal,
SNDR ≈SNRideal
ρ2(b)
F,(19)
indicating that the noise figure and the ADC affect the
SNDR in a multiplicative fashion. At high SNRideal, in
turn, the SNDR is bounded by the SDR as per (16).
C. Optimum Quantization
For a complex Gaussian signal, the quantization distor-
tion is minimized, in the mean-square sense, by a vector
quantizer operating over asymptotically long blocks and
having itself a complex Gaussian codebook [21]. Then,
the distortion is itself Gaussian and independent of the
quantized signal, and
Q(u) = p1−2−2bp1−2−2bu+d(20)
with d∼ NC0,2−2b; this relationship evinces a loss
in signal energy and the appearance of the quantization
distortion d, which in this case does amount to additional
Gaussian noise. Note that, with vector quantization over
asymptotically long blocks, bembodies the average num-
ber of bits per symbol, which need not be integer. The
number of bits Mb representing a block of Msymbols is
to be an integer, but, for M→ ∞, that allows bto take
any rational value. From (17) and (20),
ρ2(b)=1−2−2b,(21)
which plugged into (13) gives the SNDR. Moreover, with
the distortion introduced by the optimum vector quantizer
being complex Gaussian and independent of the signal,
the bound in (10) is then the actual spectral efficiency
achievable with complex Gaussian signaling, even without
a post-RFE unitary transformation. However, the situation
is not akin to an AWGN setting because, for a given
resolution, the SNDR is curbed and thus caution must
be exercised when applying known results, in particular
notions such as the degrees-of-freedom that are inherently
asymptotic.
Interestingly, the optimum vector quantizer turns out to
be a remarkably faithful representation of the scalar uni-
form quantizers that are preferred from an implementation
SDR [dB]
0
20
30
10
40
2 3 61 4 5 7
b
Optimum vector
Scalar
Approx.
Fig. 5: Noiseless SDR as a function of the ADC resolution: optimum
vector quantization vs scalar uniform quantization and its asymptotic
approximation in (23).
Power Consumption [mW]
0
10
20
30
40
50
0 2.5 5 7.5 10 12.5 15
SNRideal [dB] −SNDR [dB]
SNRideal =0,10,20,30 dB
Fig. 6: Minimum power consumption over all RFE configurations as a
function of the dB-difference between SNRideal and SNDR for fc= 28
GHz, B= 400 MHz, and scalar uniform quantization.
standpoint, with binteger-valued. Although, in general,
the ensuing ρ2(b)has to be determined numerically, for
growing b[22]
ρ2(b)≈1−c b 2−2b(22)
for some constant c. Consequently,
SNDR ≈(1 −c b 2−2b)SNRideal
F+c b 2−2bSNRideal
.(23)
Presented in Fig. 5 is a comparison between the noiseless
SDR with optimum vector quantization and with scalar
uniform quantization; also shown is how (23) matches the
latter for b≥3, requiring only the calibration of c. When
the backoff is sufficient and the distortion is introduced
solely by the ADC, information-theoretic analyses are
thereby enabled with only a correction for b≤2.
IV. DESIGN GUIDELINES:ACAS E STU DY
To make the analysis concrete, consider an exemplary
mmWave system with fc= 28 GHz and B= 400 MHz,
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6
Power Consumption [mW]
SNRideal = 0 dB
SNDR [dB]
SNR
ideal
= 10 dB
SNDR [dB]
SNR
ideal
= 30 dB
SNDR [dB]
Total
Noise figure
ADC
Saturation
10
10
1
10
10
-2
-1
2
-2 -4 -6 -8 0 2 4 6 8 5 10 15 20 25
Fig. 7: Minimum power consumption as a function of the SNDR, parameterized by SNRideal, with fc= 28 GHz, B= 400 MHz, and scalar
uniform quantization.
corresponding to B/fc= 1/70. The behavior of the
SNDR in this case is presented in Fig. 6, which shows,
for every value (referenced to SNRideal), the minimum
power consumption over all combinations of (F, Pmax, b).
The picture vividly illustrates the soaring power required
to bring the SNDR ever closer to SNRideal, even in the
best possible configuration of the RFE. Also noteworthy
is how the power consumption shifts up with SNRideal, as
better—hence more costly in terms of power—noise fig-
ures, saturation levels, and resolutions can be capitalized
on. This is a manifestation of the intricate relationship
between operating point and RFE knobs that is explored
in this section, as an illustration of how the RFE model
and the performance measures derived from it can be put
to use.
Fig. 7 breaks the RFE power consumption down into
its three components for the optimum combination of
(F, Pmax , b)at every SNDR. The values increase in dis-
crete steps, reflecting how (F, Pmax, b)have been dis-
cretized to search for the optimum combinations. Save
at the highest SNRideal, the saturation power is negligible
relative to the other components. Some caution is in order
concerning this, as more work is needed to finely model
the behavior of mixers at very low power levels, and
also because the present analysis considers a single link.
Additional linear range might have to be budgeted to
handle interference, a point that we return to later in the
article. Nevertheless, under the premise that the saturation
power can indeed be neglected, (11) simplifies to
ERFE
b=1
log2(1 + SNDR)γNF
F−1
fc
B+γADC2b,(24)
which is used in the sequel.
A. Noise Figure
Also evidenced by Fig. 7 is that, when the RFE is
optimally configured, the strongest component of the
power consumption is virtually always the one associated
with the noise figure; physically, this power is invested
in improving the quality of the LNA. Under this premise,
it follows from an inspection of (24) that, while relaxing
the noise figure worsens the spectral efficiency, it lowers
the power consumption even faster, improving the energy
per bit. This tradeoff holds up to the point, that depends
on the resolution, where the noise-figure power ceases to
be dominant; further relaxing the noise figure becomes
pointless. This is illustrated in Fig. 8, for fc/B = 70
(corresponding for instance to 400 MHz at 28 GHz, or to
2GHz at 140 GHz), with the SNDR evaluated numerically
for 6-bit scalar uniform quantization. The RFE energy per
bit is minimized by F= 2.6,3.5, and 6.5dB, respectively
for SNRideal = 0,10, and 30 dB. Noise figures above these
values yield a simultaneously lower spectral efficiency and
energy per bit; below these values, a tradeoff unfolds.
Also interesting in Fig. 8 is that a higher SNRideal
improves both the spectral efficiency and the RFE energy
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7
102 100 400
0
4
2
6
8
Eb[pJ]
Spectral Efficiency [b/s/Hz]
SNRideal = 0 dB
SNR
ideal
= 10 dB
SNR
ideal
= 30 dB
F=2.6 dB
F=3.5 dB
F=6.5 dB
RFE
Fig. 8: Spectral efficiency vs. RFE energy per bit for SNRideal = 0,10,
and 30 dB, with 6-bit scalar uniform quantization and fc/B = 70. For
each SNRideal, the noise figure is varied and the value minimizing Eb
is indicated.
per bit. In contrast, the radiated energy per bit worsens
with a higher SNRideal because of the concavity of the bit
rate as a function thereof [20, Sec. 4.1]. Although beyond
the scope of this article, we note that a tension would
arise in any holistic optimization of the operating point
that involved both transmitter and receiver.
B. ADC Resolution
Homing in on the ADC, consider first a high SNRideal.
While increasing the resolution improves the spectral
efficiency, the behavior of the RFE energy per bit is more
nuanced, and three regimes arise:
1) For small and moderate resolutions, the power con-
sumption is dominated by the noise figure term,
hence the energy per bit actually shrinks as the
resolution grows and more bits are pushed through.
In this regime, it is pointless not to increase the
resolution.
2) At some point, given its exponential dependence on
the resolution, the ADC consumption becomes pre-
dominant and the energy per bit starts moving north,
setting up a tradeoff with the spectral efficiency.
3) Eventually, the spectral efficiency ceases to improve
as the performance becomes limited by noise rather
than quantization. Past this resolution, energy is
squandered for no significant improvement in spec-
tral efficiency.
Operation in the first and last regimes is ill-advised, and
a well-designed system should target the (rather narrow)
intermediate one. This insight would be missed if only the
ADC power consumption were accounted for, rather than
that of the entire RFE, as the conclusion would then be
that the energy per bit is minimized as the resolution is
minimized. A more refined analysis reveals that interme-
diate resolutions are actually preferable. In particular, the
resolution that minimizes the energy per bit is obtainable
by solving the transcendental equation that emerges from
the condition ∂Eb
∂b = 0. For SNRideal = 30 dB, fc/B = 70,
and F= 5 dB, this returns b= 4. Shown in Fig. 9 is how
operation below this resolution is indeed unwise while
Eb[pJ]
10 1001
0
4
2
6
8
Spectral Efficiency [b/s/Hz]
SNRideal = 0 dB
SNR
ideal
= 30 dB
b=1
b=1
b=3
b=7
b=4
b= 10
SNR
ideal
= 10 dB
b=2
RFE
Fig. 9: Spectral efficiency vs. RFE energy per bit for SNRideal = 0,10,
and 30 dB with F= 5 dB and fc/B = 70. For each SNRideal, the
resolution of the scalar uniform quantizer is varied starting at b= 1 and
the value that minimizes Ebis indicated.
operation above b= 7 is rather pointless; for 4≤b≤7,
roughly a four-fold factor in energy per bit can be traded
for a roughly 40% increase in spectral efficiency.
At lower SNRideal, the same insight holds, only at
lower resolutions. For the example in Fig. 9 translated to
SNRideal = 10 dB, a 3-bit resolution minimizes the energy
per bit. This suggests that an SNR-adaptive resolution
would be desirable, and such adaptation is a challenge
worth posing to device engineers. Short of that, the
resolution must be based on the interval that a system
is meant to operate on.
Altogether, the interest in signaling strategies and re-
ceiver architectures that are tailored to coarse quantization
is well justified [3], [4], not only to prevent excessive
power consumption at the receiver, but for the sake
of efficiency. In particular, attention should be paid to
medium-resolution converters.
V. MULTIANTENNA RECEIVERS
Continuing with the case study in the previous section,
let us now graduate to multiantenna receivers, progressing
from the optimum solution at low SNR (beamforming)
to the optimum solution at high SNR (multiple-input
multiple-output, or MIMO).
A. Digital Beamforming
Soaring frequencies and bandwidths sink SNRideal, be-
cause the omnidirectional pathloss scales with f2
cand the
noise power scales with B. Beamforming is the antidote
that can bring things back to the operating range of interest
without the need to dramatically shrink the communica-
tion distance and/or increase the transmit power.
Digital beamforming at baseband requires an RFE per
antenna, followed by a maximal-ratio combiner. The exact
expression for the ensuing SNDR is somewhat involved
[23], but, reasonably regarding the distortion at different
antennas as uncorrelated, it equals the single-antenna
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8
SNDR multiplied by the number of antennas, N. With
that, (16) becomes
SNDR ≤Nρ2(b, ν)
1−ρ2(b, ν).(25)
Likewise, the power consumption would in principle be
multiplied by N. As mentioned though, the real ap-
peal of beamforming is that, as the SNDR is boosted,
proportionally lower SNRideal can be supported. Hence,
each constituent RFE can operate at a lower resolution,
with a lower power consumption. Weighing the plunging
SNRideal against the increasing beamforming gain, and the
growing number of RFEs against a lower per-RFE power
consumption, looms as a most interesting exercise.
B. Analog Beamforming
To skirt the need for a full RFE per antenna, some of the
beamforming operations can be conducted in the analog
domain, and the resulting architectures are indeed widely
used in 5G millimeter-wave systems. The signal received
by each antenna is passed through an LNA and a phase
shifter. The resulting cophased signals are then added and
downconverted, and the ensuing single stream is digitized.
Proceeding as in the single-antenna case, only with Erand
SNRideal multiplied by N,
SNDR =NSNRideal ρ2(b, ν)
F+NSNRideal (1 −ρ2(b, ν)) ,(26)
where, rather than (15), the saturation backoff is now
ν=Emax
NEr+N0
.(27)
Note that
SNDR ≤ρ2(b, ν)
1−ρ2(b, ν),(28)
which, unlike (25), is unaffected by N. It follows that a
higher resolution and saturation level are needed for the
same noiseless SDR as with digital beamforming. Indeed,
as the ADC and mixer are shared by all Nchannels, a
higher resolution and saturation power are to be expected.
As of the RFE power consumption, it abides by
PRFE =γNF
NfcG
F−1+γmaxPmax +γADC B2b,(29)
where Gis an additional LNA gain required overcome
the insertion loss of the phase shifter. At millimeter-wave
frequencies, a typical value is G= 10 dB [24]. As only the
LNA is replicated Ntimes, only the noise-figure term is
affected by the number of antennas, yet this term is now
further multiplied by the insertion loss being corrected.
And, as mentioned, a higher resolution is needed, meaning
that the ADC term is also indirectly enlarged.
A comparison of the power consumption, digital versus
analog, is presented in Fig. 10, for N= 16 antennas and
with the configuration of (F, Pmax , b)optimized at every
point. In both cases, having SNDR approach NSNRideal
(in this case 12 dB) entails an escalating power consump-
tion that reaches levels unaffordable for many mobile
102
103
Power Consumption [mW]
104
6 7 8 9 10 11 12
SNDR [dB]
Analog
Digital
Fig. 10: SNDR after beamforming as a function of the power consump-
tion with N= 16 receive antennas at fc= 28 GHz, B= 400 MHz,
and SNRideal = 0 dB: digital vs analog beamforming.
devices; this exposes the challenge of reaping the full
potential of beamforming with 16 antennas, let alone with
the even higher numbers envisioned for 6G.
An important additional conclusion of this analysis is
that the savings in ADC power in the analog structure is
more than offset by the increase in LNA power, such that
the digital option ends up consuming substantially less
power at any given SNDR. This observation is consistent
with studies such as [24], yet, as advanced, some caution
is in order. As pointed out earlier, the analysis herein
considers that the saturation power component can be
made arbitrarily low, which would require driving the
mixer at very low powers. While designs such as [2] have
used very low power mixers to sacrifice saturation for
power consumption, further work is needed to confirm that
such highly aggressive power savings can be realized.
Digital beamforming is also preferable from a per-
formance point of view as it enables communication in
multiple directions simultaneously; this enables spatial
multiplexing as well as dramatically faster beam search.
Also, digital beamforming can mitigate frequency selec-
tivity in the channel, hence its appeal lies beyond the pure
spectrum-energy tension. Our derivations suggest that,
moving beyond current commercial design choices that
employ analog beamforming to save power, the benefits of
digital beamforming could eventually come with a power
benefit.
C. MIMO
Finally turning to MIMO, which, like digital beam-
forming, necessitates of a full RFE per antenna, it can be
abstracted by the combination of a multiplexing gain and a
beamforming gain. There are some caveats concerning the
dependence on the channel-state information available at
the transmitter, and the analysis could branch accordingly,
but the multiplexing gain is robust in that respect. More
nuanced is the dependence on the channel matrix itself,
whose singular value spread determines the balance be-
tween multiplexing gain and beamforming gain. Having
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9
already studied the latter, it is sensible to now focus on
situations where the former is maximized; this befits either
rich multipath channels at lower frequencies or near-field
channels at higher frequencies [25]. In these situations,
at a fixed SNRideal both the spectral efficiency and the
consumed power scale with the number of antennas, and
thus the energy per bit is held constant. Alternatively,
with a diminishing SNRideal, the spectral efficiency can
be held constant while the energy per bit shrinks. Both
possibilities are decidedly attractive.
VI. 1-BIT COMMUNICATION
Although, as seen earlier, very low resolutions are not
the best choice in terms of energy per bit, and are the
worst choice in terms of spectral efficiency, the 1-bit case
stands apart for two reasons:
•It brings the RFE power consumption down to its
minimum, which may be enticing for devices limited,
besides battery storage, by sheer power consumption
as a proxy for cost, form factor, and heat dissipation.
•This extreme approach, in conjunction with 1-bit res-
olution also at the transmitter, is implementationally
attractive: nonlinear power amplifiers can be tolerated
at both ends—with the caveat of spectral regrowth
and adjacent channel interference—as only the signs
of the in-phase and quadrature components matter.
Note that dropping the quadrature component could be
seen as an even further reduction in resolution, but in
actuality it would sacrifice half the bandwidth, hence it is
information-theoretically sound to preserve both complex
dimensions.
With 1-bit resolutions at the two ends of the link, the
single-antenna spectral efficiency is [26]
C= 2 h1− HbQ(−√SNR)i (30)
where Hb(·)is the binary entropy function and Q(·)is
the Gaussian Q-function. Meanwhile,
ERFE
b=1
CγNF
F−1
fc
B+γADC2b(31)
where, since the spectral efficiency ceases to improve
beyond SNR ≈7dB, the noise figure can be relaxed.
The price of the superior energy per bit is, of course,
that C≤2b/s/Hz. This is compounded by the difficulties
in engineering a link with such strong nonlinearity early in
the processing [27]; in that sense, 1-bit communication is a
rather unchartered territory, with its own set of challenges.
Beamforming does not remove the 2-b/s/Hz ceiling,
although it does lessen the SNR required to attain a certain
spectral efficiency. Analog beamforming, in particular,
can be straightforwardly subsumed by using NSNR in
lieu of SNR. Digital beamforming, alternatively, requires
a generalization of (30), depending on the manner in
which the signals are combined; from an information-
theoretic perspective, the optimum manner is the one that
maximizes the mutual information [28].
MIMO, if feasible, is the clearest way to higher spectral
efficiencies in a 1-bit architecture, and the RFE count is
no different from that of digital beamforming: oa full RFE
per receive antenna. Again, to the extent that the spectral
efficiency scales with the number of antennas, the energy
per bit does not worsen, making 1-bit MIMO an attractive
combination.
Alternatively to MIMO, there are intriguing
information-theoretic results indicating that oversampling
can make up for shortages in resolution. This is the
case if the continuous-time signal is first hard-limited,
discriminating it to two levels, and then sampled. The
hard-limited signal is not bandlimited, hence sampling
it at the Nyquist rate is suboptimal [29]. For a properly
crafted transmit signal distribution, the 2-b/s/Hz ceiling
can be raised logarithmically in the oversampling factor
[30]. Likewise, if the signal is sampled and subsequently
1-bit quantized, oversampling can be exploited provided
the receiver lowpass filter is not set to the signal
bandwidth, but expanded by the oversampling factor.
Although that increases the noise power, it also ensures
that the various samples corresponding to each symbol
are contaminated by independent noise and, again for
carefully designed signal distributions, a ceiling above
2-b/s/Hz ceiling can then be attained [31]. Altogether,
oversampling can be an interesting alternative to 1-bit
communication whenever MIMO is not an option.
No oversampling has been posited throughout this
article, and the bandwidth has been identified with the
sampling rate. If that is not the case, then the sampling
rate should be plugged in lieu of the bandwidth in the
RFE power consumption formulas.
VII. CONCLUSION
As strides are taken towards 6G, energy efficiency is
poised to regain its importance alongside spectral ef-
ficiency. To provide information-theoretic cover to this
development, classical formulations should be augmented
with the power consumed by the circuitry. At mobile
devices in particular, that consumption is to be dominated
by the RFE as we move to higher frequencies, bandwidths,
and antenna counts.
Instrumental to any analysis involving the RFE is a
model for its power consumption, and this article has
set forth such a model. Then, following in the foot-
steps of other works, the spectral efficiency has been
expressed with explicit account of the added noise and
distortion introduced by the RFE. The tradeoff between
such spectral efficiency and the energy per bit sheds
light on the effect of the operating point and the FRE
configuration. As a case study, receivers with escalating
frequencies and bandwidths have been contemplated, and
the gleaned insights include the optimality of intermediate
(SNR-dependent) resolutions. The 12-bit or even 14-bit
resolutions employed by 4G and 5G become decidedly
inadequate, and suitably adapting the resolution to the
SNR appears as desirable should the hardware be able to
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10
accommodate it. The machinery put forth in the article
could be put to use for other situations of interest to
6G. Particularly, multiuser channels with strong near-
far conditions as well as adversarial systems would de-
mand a much higher saturation backoff, bringing into the
picture additional distortion and/or power consumption,
and possibly altering the contrast of digital and analog
beamforming. The corresponding analysis would be a
welcome development.
In closing, some final comments are in order:
•In the face of the non-Gaussian distortion generated
by scalar uniform quantizers, and possibly by signal
saturation, Gaussian signaling ceases to be optimum.
In particular, as far as the quantization distortion
is concerned, discrete signal distributions have been
shown to be optimum [3]. With optimized discrete
constellations, therefore, somewhat better spectral
efficiency and energy per bit than the ones considered
in this article are possible, yet that need not be the
case with generic constellations. Ongoing research
on this matter could augment the framework herein
presented.
•Interference would exhibit a more favorable behavior
than thermal noise in that its spectral density would
abate as the bandwidth of the interfering transmitters
grows; ultimately, the large-bandwidth regime is in-
herently not limited by interference, but by noise and
distortion.
•Fading and the acquisition of channel-state infor-
mation are problems on their own right, and they
would affect the spectral-energy tradeoff as the band-
width grows unboundedly—provided the scattering
were endlessly rich. However, multipath propagation
becomes decidedly sparser as the frequency grows
into the realm where very large bandwidths are
possible, eroding the inherent channel uncertainty
and its impact.
A host of research avenues open up as energy effi-
ciency, device power consumption, coarse quantization,
distortion, oversampling, and related aspects are injected
into information-theoretic formulations. Opportunities to
continue probing the boundaries and guiding the evolution
of wireless systems, and in particular to keep closing the
knowledge gap in terms of how much energy is funda-
mentally needed to reliably convey one bit of information
across a wireless channel.
ACKNOWLEDGMENT
The feedback provided by the guest editorial team and
the anonymous reviewers is gratefully acknowledged.
REFERENCES
[1] S. Wesemann, J. Du, and H. Viswanathan, “Energy efficient
extreme MIMO: Design goals and directions,” IEEE Commun.
Magazine, vol. 61, 2023.
[2] P. Skrimponis, N. Hosseinzadeh, A. Khalili, E. Erkip, M. J. Rod-
well, J. F. Buckwalter, and S. Rangan, “Towards energy efficient
mobile wireless receivers above 100 GHz,” IEEE Access, vol. 9,
pp. 20 704–20 716, 2020.
[3] J. Singh, O. Dabeer, and U. Madhow, “On the limits of com-
munication with low-precision analog-to-digital conversion at the
receiver,” IEEE Trans. Commun., vol. 57, no. 12, pp. 3629–3639,
2009.
[4] A. Mezghani and J. A. Nossek, “Capacity lower bound of MIMO
channels with output quantization and correlated noise,” in IEEE
Int’l Symp. Inform. Theory (ISIT), 2012.
[5] J. N. Murdock and T. S. Rappaport, “Consumption factor and
power-efficiency factor: A theory for evaluating the energy effi-
ciency of cascaded communication systems,” IEEE J. Sel. Areas
Commun., vol. 32, no. 2, pp. 221–236, 2013.
[6] O. Kanhere et al., “A power efficiency metric for comparing energy
consumption in future wireless networks in the millimeter-wave
and terahertz bands,” IEEE Wireless Commun., vol. 29, no. 6, pp.
56–63, 2022.
[7] A. Lozano and S. Rangan, “Spectral vs. Energy Efficiency in 6G:
Impact of the Receiver Front-End,” arxiv preprint, 2023.
[8] I. Song, J. Jeon, H.-S. Jhon, J. Kim, B.-G. Park, J. D. Lee, and
H. Shin, “A simple figure of merit of RF MOSFET for low-noise
amplifier design,” IEEE Electron Device Letters, vol. 29, no. 12,
pp. 1380–1382, 2008.
[9] L. Belostotski and E. A. Klumperink, “Figures of merit for CMOS
low-noise amplifiers and estimates for their theoretical limits,”
IEEE Trans. Circuits and Systems II: Express Briefs, vol. 69, no. 3,
pp. 734–738, 2021.
[10] R. Brederlow, W. Weber, J. Sauerer, S. Donnay, P. Wambacq, and
M. Vertregt, “A mixed-signal design roadmap,” IEEE Design &
Test of Computers, vol. 18, no. 6, pp. 34–46, 2001.
[11] R. Ramzan, F. Zafar, S. Arshad, and Q. Wahab, “Figure of merit
for narrowband, wideband and multiband lnas,” Int’l Journal of
Electronics, vol. 99, no. 11, pp. 1603–1610, 2012.
[12] H.-k. Chen, J. Sha, D.-c. Chang, Y.-Z. Juang, and C.-f. Chiu,
“Design trade-offs for low-power and high figure-of-merit LNA,”
in Int’l Symp. VLSI Design, Automation and Test, 2006, pp. 1–4.
[13] A. Verma, P. K. Yadav, S. Ambulker, M. Goswami, and P. K. Misra,
“A 36.7 mW, 28 GHz receiver frontend using 40 nm RFCMOS
technology with improved figure of merit,” Analog Integrated
Circuits and Signal Processing, vol. 107, pp. 135–144, 2021.
[14] B. Murmann, “The race for the extra decibel: A brief review of
current ADC performance trajectories,” IEEE Solid-State Circuits
Magazine, vol. 7, no. 3, pp. 58–66, 2015.
[15] C.-Y. Wang and J.-H. Tsai, “A 51 to 65 ghz low-power bulk-driven
mixer using 0.13 µm CMOS technology,” IEEE Microwave and
Wireless Components Letters, vol. 19, no. 8, pp. 521–523, 2009.
[16] B. Guo, H. Wang, and G. Yang, “A wideband merged CMOS active
mixer exploiting noise cancellation and linearity enhancement,”
IEEE Trans. Microwave Theory and Techniques, vol. 62, no. 9,
pp. 2084–2091, 2014.
[17] S. Dutta, A. Khalili, E. Erkip, and S. Rangan, “Capacity bounds for
communication systems with quantization and spectral constraints,”
in IEEE Int’l Symp. Inform. Theory (ISIT), 2020, pp. 2038–2043.
[18] A. Lapidoth, “Nearest neighbor decoding for additive non-Gaussian
noise channels,” IEEE Trans. Inform. Theory, vol. 42, no. 5, pp.
1520–1529, 1996.
[19] A. Lapidoth and P. Narayan, “Reliable communication under
channel uncertainty,” IEEE Trans. Inform. Theory, vol. 44, no. 6,
pp. 2148–2177, 1998.
[20] R. W. Heath Jr. and A. Lozano, Foundations of MIMO Communi-
cation. Cambridge University Press, 2018.
[21] A. Gersho and R. Gray, Vector quantization and signal compres-
sion. Springer Science & Business Media, 2012, vol. 159.
[22] D. Hui and D. L. Neuhoff, “Asymptotic analysis of optimal fixed-
rate uniform scalar quantization,” IEEE Trans. Inform. Theory,
vol. 47, no. 3, pp. 957–977, 2001.
[23] A. Khalili, E. Erkip, and S. Rangan, “Quantized MIMO: Channel
capacity and spectrospatial power distribution,” in IEEE Int’l Symp.
Inform. Theory (ISIT), 2022, pp. 2303–2308.
[24] S. Dutta, C. N. Barati, D. Ramirez, A. Dhananjay, J. F. Buckwalter,
and S. Rangan, “A case for digital beamforming at mmWave,”
IEEE Trans. Wireless Commun., vol. 19, no. 2, pp. 756–770, 2019.
[25] H. Do, S. Cho, J. Park, H.-J. Song, N. Lee, and A. Lozano, “Ter-
ahertz line-of-sight MIMO communication: Theory and practical
challenges,” IEEE Commun. Magazine, vol. 59, no. 3, pp. 104–109,
2021.
[26] A. J. Viterbi and J. K. Omura, “Principles of digital communication
and coding,” 1979.
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content may change prior to final publication. Citation information: DOI 10.1109/MBITS.2023.3322975
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11
TABLE II: Theoretical bit rate and power consumption for a short range,
low power, high rate application.
Parameter Value Remarks
Transmit power 1 mW
fc3.5 GHz Common in private 5G
B200 MHz
Path loss model 3GPP InH Indoor office
Distance 20 m
SNRideal 10.3 dB
F4 dB
b4 bits/dim
ν30 dB
Spectral efficiency 0.8 log2(1 + 0.25 SNDR)[bps/Hz]
Calculated Values for the Bit Rate
SNRideal 10.3 dB
SNDR 6.0 dB
R160 Mbps
Calculated Values for the RFE Power
Saturation power 0.003 mW γmax Pmax
ADC power 0.52 mW γADCB2b
NF power 0.32 mW γNF fc
F−1
PRFE 0.84 mW
[27] J. Singh, S. Ponnuru, and U. Madhow, “Multi-gigabit communica-
tion: The ADC bottleneck,” in IEEE Int’l Conf. Ubiquitous Wireless
Broadband (ICUWB), 2009, pp. 22–27.
[28] K. Gao, J. N. Laneman, and B. Hochwald, “Beamforming with
multiple one-bit wireless transceivers,” in Inform. Theory and
Applic. Workshop (ITA), 2018.
[29] E. N. Gilbert, “Increased information rate by oversampling,” IEEE
Trans. Inform. Theory, vol. 39, no. 6, pp. 1973–1976, 1993.
[30] S. Shamai, “Information rates by oversampling the sign of a
bandlimited process,” IEEE Trans. Inform. Theory, vol. 40, no. 4,
pp. 1230–1236, 1994.
[31] S. Krone and G. Fettweis, “Capacity of communications channels
with 1-bit quantization and oversampling at the receiver,” in 35th
IEEE Sarnoff Symposium, 2012.
APPENDIX A
LOW-P OWER, HIGH BIT RATE USE CA SE
An important implication of the analysis herein is
the potential for a high bit rate, short range, but very
low power wireless system. As an example, consider the
parameters in Table II, with a relatively low transmit
power of 1 mW (similar to Bluetooth low energy). The
frequency and bandwidth are consistent with private 5G
networks in the CBRS band. With these parameters,
SNRideal = 10.3 dB. Numerically computing ρ2(b, ν)and
applying (13), we obtain SNDR ≈6.0 dB; this is about
4.3 dB below SNRideal. To enable a low power digital
implementation, assume that only 80% of capacity at 6 dB
is attained. Then, R≈160 Mbps.
What is remarkable is that the consumption can be
small. Combining Table I with (6), the total RFE power
consumption is below 1 mW; see breakdown in Table II.
Of course, this only accounts for the RFE power, to which
the digital processing required for filtering, equalization,
and decoding, must be added. Nevertheless, this simple
calculation supports the possibility of high bit rate appli-
cations at very low powers.
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content may change prior to final publication. Citation information: DOI 10.1109/MBITS.2023.3322975
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